In On Liberty (1859), John Stuart Mill exposes an “all but universal illusion” caused by “the magical influence of custom:” even though societal norms and rules of conduct vary wildly between different ages and countries, the people of any given time and place always believe their own norms and rules to be self-evident and self-justifying. The root of the problem is people mistaking their strong feelings for good reasons:

The effect of custom, in preventing any misgiving respecting the rules of conduct which mankind impose on one another, is all the more complete because the subject is one on which it is not generally considered necessary that reasons should be given, either by one person to others, or by each to himself. People are accustomed to believe, and have been encouraged in the belief by some who aspire to the character of philosophers, that their feelings, on subjects of this nature, are better than reasons, and render reasons unnecessary. The practical principle which guides them to their opinions on the regulation of human conduct, is the feeling in each person’s mind that everybody should be required to act as he, and those with whom he sympathizes, would like them to act. No one, indeed, acknowledges to himself that his standard of judgment is his own liking; but an opinion on a point of conduct, not supported by reasons, can only count as one person’s preference; and if the reasons, when given, are a mere appeal to a similar preference felt by other people, it is still only many people’s liking instead of one. To an ordinary man, however, his own preference, thus supported, is not only a perfectly satisfactory reason, but the only one he generally has for any of his notions of morality, taste, or propriety, which are not expressly written in his religious creed; and his chief guide in the interpretation even of that.

This way of thinking invariably leads to a “tyranny of the majority,” even under democratic government. How should this be avoided?

The object of this Essay is to assert one very simple principle, as entitled to govern absolutely the dealings of society with the individual in the way of compulsion and control, whether the means used be physical force in the form of legal penalties, or the moral coercion of public opinion. That principle is, that the sole end for which mankind are warranted, individually or collectively, in interfering with the liberty of action of any of their number, is self-protection. That the only purpose for which power can be rightfully exercised over any member of a civilized community, against his will, is to prevent harm to others. His own good, either physical or moral, is not a sufficient warrant. He cannot rightfully be compelled to do or forbear because it will be better for him to do so, because it will make him happier, because, in the opinions of others, to do so would be wise, or even right. These are good reasons for remonstrating with him, or reasoning with him, or persuading him, or entreating him, but not for compelling him, or visiting him with any evil in case he do otherwise. To justify that, the conduct from which it is desired to deter him, must be calculated to produce evil to some one else. The only part of the conduct of any one, for which he is amenable to society, is that which concerns others. In the part which merely concerns himself, his independence is, of right, absolute. Over himself, over his own body and mind, the individual is sovereign.

Though many people today would (hopefully) agree with the above in principle, in practice it remains easy to fall into the trap of attempting to force others into doing what you think is good for them, or to prevent them from doing what you think is bad for them. One good example is marijuana use; another is pornography. In an interview with author and pornographic actress Nina Hartley, she is asked how she responds to feminists who criticize pornography as “an industry that subjects women to men’s desires.” Hartley says:

In essence… feminism means that I have choice in my life — autonomy. It’s hard to imagine now, but fifty years ago a woman couldn’t get birth control if she was single, and even married women needed their husband’s permission to get it. Back-alley abortions killed or maimed thousands of women each year. It was nearly impossible to bring abuse charges against a husband or wife. Younger people don’t realize how bad it was, or how recent. Feminists of the 1960s and ’70s really busted the door wide open and we’re still sorting it all out. We take for granted now that spousal abuse is a crime but it wasn’t like that then. Women also fought hard for the right to be sexual on our own terms (having sex before marriage, not becoming mothers if we didn’t want to), and really made a lot of headway in raising consciousness surrounding rape, which is now taken seriously. A woman’s sexual history can no longer be used against her in court when she faces her attacker.

In a nutshell: my body, my rules. Other women don’t get to tell me what’s “right” for me, just as no imam, rabbi, priest or minister gets to tell me what to do with my body. You don’t know better than I do what I need, so don’t presume to.

But surely, no woman would ever choose to be a pornographic actress of her own free will! Surely?

Whether or not we agree with or approve of them, the choices made by young women are theirs. If we’re to grant autonomy to people over the age of eighteen, then that means accepting their choices as valid, even if we’d never do such a thing. This includes being able to join the army and get shot or maimed, or become a miner or construction worker. Those are deadly jobs (no one has died from making porn in the thirty-seven years it’s been legal) and no one thinks to tell a young adult, “Don’t do that job, it’s dangerous.” Or if we do tell them, we accept that, being young people, they may disregard our advice. If we accept that a young woman can consent to have an abortion or become a parent, then it stands to reason that we must accept that she can consent to make pornography. . . .

The widespread notion that legal porn production is a sink hole of abuse and coercion that takes advantage of poor, innocent women, is the biggest smack leveled against the business. It’s almost entirely a function or projection of people’s fears and discomfort about women, gender relations, sex, sexuality and the graphic depiction of sexual acts. The idea that a woman could choose, on purpose, to perform in pornographic videos for her own reasons still goes deeply against the notion that women are somehow victims of male sexuality, that they’re delicate flowers who need the protection of a good man, or the law.

The only protection that the law needs to provide us is protection from others — not from ourselves. Mill famously put it this way:

The only freedom which deserves the name, is that of pursuing our own good in our own way, so long as we do not attempt to deprive others of theirs, or impede their efforts to obtain it. Each is the proper guardian of his own health, whether bodily, or mental and spiritual. Mankind are greater gainers by suffering each other to live as seems good to themselves, than by compelling each to live as seems good to the rest.

Five children have been playing together, and three of them have gotten mud on their foreheads. Each child can see mud on others but not on himself. When the children come home, their father says that at least one of them has mud on his forehead; he then asks if anyone can deduce whether there is mud on his own forehead. The children look around, but no one answers. So the father asks again: Does anyone know whether he has mud on his own forehead? Silence. The father then repeats his question a third time, at which point all three dirty children immediately step forward and proclaim that their foreheads must be muddy.

The first (and simplest) puzzle is: How did the kids know? And why did the father have to ask three times?

The solution is inductive: begin by considering what would happen if there were only one muddy child, Alex. He would see that all the others are clean, so when the father states that at least one child is dirty, Alex would immediately know it must be him. Therefore, if there were only one dirty child, he would come forward after the father’s first query. Next, suppose there are two muddy children, Alex and Bob. Alex sees that Bob is dirty, and vice versa. As far as Alex knows, Bob could be the only dirty one, and as far as Bob knows, Alex could be the only dirty one; so neither of them step forward after the father’s first query (each expecting the other to do so). However, when Alex sees that Bob did not immediately come forward, and since we already concluded that if there were only one muddy child he would have identified himself right away, Alex can deduce that his own forehead must be muddy as well. Bob reasons likewise, and so they both step forward when the father asks a second time. Applying the same logic to our original scenario shows that the three dirty children can identify themselves as soon as they see that no one came forward after the second query — so in this case, the third time is the charm.

A more interesting puzzle, however, is this: Was the father’s opening statement that “at least one child has mud on his forehead” necessary? Reexamining the solution above, we find that the “base case” in which there is only one muddy child doesn’t work without the father’s statement — the dirty child would never come forward, since as far as he knows, no one is muddy. Consequently, the entire chain of inference collapses. So the father’s statement is necessary — but this seems paradoxical, since he merely told the children something they already know! After all, if there are three muddy foreheads, each of the five children knows just by looking at the others that “at least one child has mud on his forehead.” So what information does the father’s statement actually provide?

Think again about the case where only two children have mud on their foreheads, Alex and Bob, and let p represent the statement “at least one child has mud on his forehead.” It is then true that each of the five children knows p even without the father saying so, but the problem is that not everyone knows that everyone else knows p: as far as Alex knows, Bob could be the only muddy one, in which case Bob would not know that “at least one child is muddy” (since Bob would see only clean faces). Likewise, as far as Bob knows, Alex might not know p. This is what prevents them from deducing their own situation. What about our original case? When there are three dirty children, not only does everyone already know p, everyone knows that everyone knows p. However, not everyone knows that everyone knows that everyone knows p! What the father provides, then, is common knowledge: after his statement, they all know that there is at least one muddy child, and they all know that they all know it, and they all know that they all know that they all know it…

A closely related scenario is the coordinated attack problem. Suppose that two generals, each in command of an army division, wish to launch a surprise attack on the enemy. However, the only way for the attack to succeed is if both divisions attack simultaneously; if a single division attacks alone it will be defeated. Unfortunately, the only way for the generals to communicate is by messenger, and the messengers can be lost or delayed along the way. What should the generals do? Suppose that general A sends general B the message: “Attack at dawn on January 1st.” This is certainly not enough to guarantee a coordinated attack, since general A cannot be sure that his message was received. So let’s say that general A waits until he receives a confirmation message back from general B. Can both generals then attack with confidence? No, since general B doesn’t know whether the confirmation he sent was received by general A — and general B knows that general A will not attack without receiving that confirmation. (And general A knows that general B knows this.) So even though both generals know when to attack, and they each know that the other knows, they cannot attack because neither general knows that the other knows that he knows!

This situation is familiar to anyone trying to make an appointment with someone by email. How many rounds of confirmation are necessary to be sure you both know the engagement is on? It’s easy to see that no number of acknowledgments and counter-acknowledgments will allow the parties to achieve absolute certainty. As explained in a paper by Ronald Fagin, Joseph Halpern, Yoram Moses and Moshe Vardi, guaranteeing coordinated attack requires common knowledge, and common knowledge cannot be achieved where communication is not guaranteed. Actually, the situation is worse than that: even in a system where communication is guaranteed, if there is any uncertainty about the delivery time — e.g., even if all messages are guaranteed to arrive in one millisecond or less — common knowledge is impossible to achieve.

Fagin et al. go on to explore various ways out of this apparent stalemate. But if you ever miss a date, you can always claim that you didn’t show up because the other party didn’t acknowledge your acknowledgment of their acknowledgment…

When considering the recent religious ruling by dozens of municipal chief rabbis in Israel, forbidding Jews from renting or selling their houses to non-Jews, Daniel Gordis opines that concepts like “racism,” while relevant in America, may not be applicable to Israel:

[…] America does not need to struggle to guarantee its Christian nature. Our society, though largely Jewish now, could easily become something very different with time. If that is what these rabbis meant to say, they were right.

Apply the ethnicity-blind standards of American life here, and in a generation or two, Israel’s Jewish quality might be gone.

Gordis goes on to imply that discriminating against non-Jews may be acceptable in order to “guarantee a long-term Jewish quality of this country”. To see the problem with this approach, begin by noticing that there is not actually any agreement among Israeli Jews about just what kind of “Jewish quality” Israel ought to have. There is not even any agreement about who is considered a Jew in the first place — witness the never-ending political struggles over conversion. Would it be acceptable to discriminate against Conservative and Reform Jews in order to guarantee a long-term Orthodox quality for Israel? (“I won’t rent my house to Conservatives!”) Conversely, many Israeli Jews consider the Haredi way of life to be antithetical to the kind of state they would like to live in; would we be justified in treating Haredim differently from all other citizens? (“Haredim are not welcome in my store!”)

Taking Gordis’s approach to its logical conclusion would seem to entail the creation of separate countries for every different culture/worldview. But what happens when people who have lived their whole lives in a Jewish (Orthodox?) state decide they want to be Christians (or Reform Jews) — must they leave their homes and move to a Christian (or Reform) country? Or face discrimination if they stay? And how exactly are we going to decide who is considered a “real Jew” for this purpose? For that matter, there’s no reason why religion should be the only kind of “quality” a state might want to preserve — would the French be justified in discriminating against citizens who want to change their country’s wine and cheese culture?

The fact is that we will always have to live in a society together with people who have different opinions and priorities, and that is actually a good thing: it exposes us to different perspectives and forces us to think critically and to rigorously defend our views in a free marketplace of ideas — may the best ideas win. Being surrounded only by like-minded people is a recipe for stagnation. And as Jews should be the first to remember, any of us could one day find ourselves in the minority. Only a system where all citizens are treated equally can guarantee that people with different beliefs and backgrounds can live together peacefully, and be free to choose their own lives. The “quality” of a state may indeed change over time, but that is often beneficial: would any of us want to live in a Biblical Jewish society, where slavery is permitted and where a woman who is raped must marry her rapist?

Each of us has values and traditions he would like to preserve; people of similar worldviews are free to get together as communities, and build themselves a thriving cultural life. But the honest way to preserve a culture is by persuading other people of its value. If the only way to keep your culture alive is by forcing it on others and discriminating against those who are different, your culture ought to go extinct.

Despite appearances, the two tabletops are identical in both size and shape.

I’m preparing a special lesson for my high school students on the subject of critical thinking and the scientific method. In the first half of the lesson, I aim to undermine the students’ certainty about what they think they know, by demonstrating the many types of errors and biases we are all prone to. I’ll start with optical illusions, like Roger Shepard’s “Turning the Tables.” Carefully measuring the two tabletops shows that they are identical in both size and shape, even though we feel very strongly that this is not the case.

Exactly one of the doors has a new car behind it. After you choose door 1, the host opens door 3 to reveal a goat. He then offers you the chance to switch to door 2. Should you?

We know we cannot always trust our senses, but surely our intuitions are better in more theoretical areas? Try this: if a bat and a ball together cost $1.10, and the bat costs $1.00 more than the ball, how much does the ball cost? Sounds easy, and it is, but a majority of university students give the wrong answer (hint: it’s not 10 cents). Our intuitions are especially bad when it comes to probability. I’ll present the famous Monty Hall problem, to which many smart people refuse to accept the correct solution (switching doors doubles your chances of winning) even after it’s explained to them. I may also mention the gambler’s fallacy — if ten consecutive tosses of a fair coin come up heads, is tails more likely to come up next? — which makes lots of money for casinos.

Instead of jumping to conclusions based on intuition, we can attempt to construct formal logical inferences: if all Greeks are handsome, and Socrates is a Greek, then Socrates is handsome. Do we ever make mistakes in our use of logic? Consider this: if some men are doctors, and some doctors are tall, can we conclude that some men are tall? (No.) Or this: if we assume that all reptiles lay eggs, and we know that alligators lay eggs, does it follow that alligators are reptiles? (Nope.)

I next turn to the fallacy of assuming that correlation implies causation. If the percentage of black people among those convicted of violent crimes is significantly greater than the percentage of blacks in the population, can we conclude that blacks are inherently more violent? No: it could be that most judges are white and some are prejudiced, or that blacks are poorer on average and poverty causes crime, or that blacks are treated as second-class citizens creating a self-fulfilling prophecy, etc. Another example: let’s say surveys show that people who use olive oil are less likely to develop heart disease than people who use other oils. Does it follow that olive oil helps prevent heart disease? Consider that olive oil is more expensive than other oils, so people who buy olive oil are more likely to belong to higher socio-economic groups — implying a more healthy diet in general, a higher chance of belonging to a gym, more money to spend on health care, etc. Of course, this does not prove that olive oil does nothing to prevent heart disease: merely that the causal connection cannot be deduced from the correlation alone.

Perhaps the most subtle causes of error are cognitive biases. I’ll talk about confirmation bias: people tend to give more weight to evidence that supports what they already believe; they tend to seek data that confirm their hypotheses instead of attempting to disprove them; they tend to remember examples that support their theories and forget those that don’t. One study was conducted on two groups of people: one group contained people who were in favor of capital punishment, and the other group contained people who were against it. All subjects were shown the same set of data, which included evidence and argument for both sides of the issue. Participants from both groups tended to report that the data had caused them to strengthen their original beliefs! So much for objectivity. Confirmation bias is also what keeps many superstitions alive: people notice those times when unlucky events happen on the 13th floor, and disregard the times when they don’t, or when they happen on other floors.

I’ll conclude the first part of the lesson with the fallacy of appealing to authority: “Einstein was a genius, so whatever he said must be so;” “If it’s written in the Bible, it must be true;” “Democracy is the best form of government, because that’s what they taught us in school;” “My teacher says that appeals to authority are logical fallacies.” The truth or falsity of a claim is not affected by the authority of the claimant; even Einstein made mistakes.

So far, then, we’ve seen that our senses can deceive us; our intuitions are often wrong; we are prone to logical fallacies and cognitive biases; it’s difficult for us to be objective; and even smart people can be mistaken. Is it impossible to obtain reliable information about the world?

Enter the scientific method. I’ll present the basic model of gathering evidence, offering a hypothesis to explain the observed phenomenon, making predictions based on the hypothesis, testing those predictions, and revising the hypothesis based on new data. This method is not infallible, of course, but science makes use of many mechanisms for minimizing errors and correcting them: transparency, documentation, reproducibility, peer review, etc.

To further explore the nature of scientific theories, I’ll use Carl Sagan’s example of the fire-breathing dragon in my garage: when my friend asks to see it, I reply that it’s invisible. She then suggests that we spread flour on the floor and look for tracks, but I explain that this dragon floats in the air. And there’s no point in trying to touch it, either, because it’s incorporeal. At this point my friend would hopefully begin to wonder what makes me think the dragon exists at all. The dragon hypothesis is unscientific because it’s unfalsifiable — there is no evidence that could possibly disprove it. This makes it useless: if there could never be any detectable difference between a world in which the dragon exists and one where it doesn’t, why should we care?

Another useful heuristic for judging scientific theories is Occam’s razor: all other things being equal, the simplest explanation of the facts is usually the right one. In other words, we should strive to minimize unnecessary or arbitrary assumptions. If I hear the clacking of hoofs coming from inside a race track, for instance, it could theoretically be a zebra escaped from the zoo, or a recording designed to fool me, or an alien language — but absent any evidence for those hypotheses, it makes sense to tentatively assume that it’s horses I’m hearing. It’s important to stress that all scientific knowledge is provisional: we can never achieve absolute certainty, but our confidence in an hypothesis grows with the amount of supporting evidence. A scientific theory is a hypothesis of sufficient explanatory power which has withstood all attempts to falsify it. But all theories are always open to revision based on new evidence.

Homology of forelimbs in mammals is evidence of evolution.

I’ll give two examples of successful scientific theories and the evidence supporting them. Firstly, how do we know the Earth is spherical? Thousands of years ago, people had already noticed that the stars in the night sky look different from different locations, that the sails of a ship can be seen on the horizon before its hull, and that the shadow cast by the Earth on the moon during a lunar eclipse is round. More recently, of course, people have circumnavigated the globe and even seen it from space. My second example is the theory of evolution, supported by evidence from fossils, comparative anatomy, the geographical distribution of plants and animals, genetics (DNA), artificial selection (e.g. dog breeding), observed natural selection (e.g. antibiotic-resistant bacteria), and more.

Let’s now try to apply the scientific method to medical testing. Does the fact that my condition improved after taking a certain drug or undergoing a certain therapy mean that the drug or the therapy are inherently effective? No: we must take into account the placebo effect, confirmation bias, the possibility of coincidence, etc. We can, however, attempt to neutralize those factors — and raise our confidence in a treatment’s effectiveness — by performing controlled double-blind trials, and analyzing the results statistically.

Having previously rejected appeals to authority, it’s important to point out the difference between authority and expertise. While the statement “90% of mathematicians recommend Acme toothpaste” does not carry any special weight, the statement “90% of dentists recommend Acme toothpaste” does. When we (provisionally!) accept the consensus of experts on matters of fact in their field of expertise, we are not doing so merely on the basis of their authority: we are relying on the scientific method itself, which includes all the self-correcting and error-minimizing mechanisms mentioned above.

The first smallpox vaccine was developed in 1796; smallpox was eradicated in 1979.

The strongest argument in favor of the scientific method is that it gets amazing results. Science can land a spacecraft on Mars, and can predict the exact time of an eclipse a thousand years in the future. Closer to home, the smallpox virus — which killed hundreds of millions of people over 10,000 years — was eradicated in 1979 after a successful vaccination campaign, the culmination of centuries of scientific effort. We often take such things for granted, but until the beginning of the 20th century, human life expectancy was only 30-40 years. It’s now around 80 years in the developed world.

This is a good place to give the students a chance to practice their own critical thinking skills on real-world examples, like the assertions of faith-healers or astrologers. What questions should we ask before accepting such claims? Are there any fallacies or biases that may be getting in the way? What data support these hypotheses, and are there more parsimonious ways of explaining them? What experiments could we perform to help us decide?

In summation: it’s essential to evaluate all ideas critically. The smartest people we know could be wrong; we should never accept something just because someone said so, or because it’s tradition, or because it feels right intuitively. Our level of confidence in a proposition ought to scale with the level of available evidence in its support. Being skeptical of extraordinary claims is a good default position — but we must also make sure to keep an open mind and be willing to consider strange and unintuitive ideas (quantum physics, anyone?). We must recognize that there are many things we do not know, and that some of what we think we know may be mistaken. It is possible, however, to expand our knowledge of the world, and to correct previous mistakes, using the scientific method.

I’ll conclude with an excerpt from Carl Sagan’s The Demon-Haunted World:

Except for children (who don’t know enough not to ask the important questions), few of us spend much time wondering why Nature is the way it is; where the Cosmos came from, or whether it was always here; if time will one day flow backward, and effects precede causes; or whether there are ultimate limits to what humans can know. There are even children, and I have met some of them, who want to know what a black hole looks like; what is the smallest piece of matter; why we remember the past and not the future; and why there is a Universe.

Every now and then, I’m lucky enough to teach a kindergarten or first-grade class. Many of these children are natural-born scientists — although heavy on the wonder side and light on scepticism. They’re curious, intellectually vigorous. Provocative and insightful questions bubble out of them. They exhibit enormous enthusiasm. I’m asked follow-up questions. They’ve never heard of the notion of a ‘dumb question’.

But when I talk to high school seniors, I find something different. They memorize ‘facts’. By and large, though, the joy of discovery, the life behind those facts, has gone out of them. They’ve lost much of the wonder, and gained very little scepticism. They’re worried about asking ‘dumb’ questions; they’re willing to accept inadequate answers; they don’t pose follow-up questions; the room is awash with sidelong glances to judge, second-by-second, the approval of their peers. . . .

There are naive questions, tedious questions, ill-phrased questions, questions put after inadequate self-criticism. But every question is a cry to understand the world. There is no such thing as a dumb question.

Bright, curious children are a national and world resource. They need to be cared for, cherished, and encouraged. But mere encouragement isn’t enough. We must also give them the essential tools to think with.

I won’t be giving the lesson for a while yet, so I’d be happy to hear any comments, criticisms and suggestions.

According to Rabbi Alan Lurie in The Huffington Post, all those debates about the existence of God — the ones where skeptics keep demanding evidence — are missing the point:

arguments and experiments can not prove the existence of God because God is not an hypothesis. For human beings, God is the experience of a transformative relationship with creation itself, in which we know that the Universe is inherently meaningful, that we were created for a staggering purpose that will unfold over eons, that love and gratitude are the essential actual materials of our lives and that we are holy beings.

The experience of a relationship with God is not one of religious doctrine, does not come from statistics, experiments or argument, and is certainly not in conflict with science and reason in any way. It is also not about righteous certainty or judgment. The experience of God expands the possibilities for our lives and increases the feeling of mystery and intellectual curiosity about the world.

I’m certainly in favor of love and gratitude and intellectual curiosity, but wait a minute: if God is merely an experience, then he cannot be an entity who writes books and listens to prayers and cares about what we eat or whom we sleep with. In what sense can Lurie call himself a theist, much less a religious Jew? Lurie is simply equivocating on the definition of “God,” emptying it of all the characteristics attributed by religious people (without which religion makes no sense), while still recommending a relationship with God (which somehow allows him to know that we were all created for some staggering purpose).

I hate to break it to you, Rabbi, but it seems that you’re stuck in a delusional relationship. You need to let go.

It is lucky that it is not windy today. Strange, how in some way one always has the impression of being fortunate, how some chance happening, perhaps infinitesimal, stops us crossing the threshold of despair and allows us to live. It is raining, but it is not windy. Or else, it is raining and is also windy: but you know that this evening it is your turn for the supplement of soup, so that even today you find the strength to reach the evening. Or it is raining, windy and you have the usual hunger, and then you think that if you really had to, if you really felt nothing in your heart but suffering and tedium — as sometimes happens, when you really seem to lie on the bottom — well, even in that case, at any moment you want you could always go and touch the electric wire-fence, or throw yourself under the shunting trains, and then it would stop raining.

In the afterword, Levi responds to common questions he received from his readers. In his answer to the question “How can the Nazis’ fanatical hatred of the Jews be explained?” he writes:

The ideas [Hitler and Mussolini] proclaimed were not always the same and were, in general, aberrant or silly or cruel. And yet they were acclaimed with hosannahs and followed to the death by millions of the faithful. We must remember that these faithful followers, among them the diligent executors of inhuman orders, were not born torturers, were not (with a few exceptions) monsters: they were ordinary men. Monsters exist, but they are too few in number to be truly dangerous. More dangerous are the common men, the functionaries ready to believe and to act without asking questions …

It is, therefore, necessary to be suspicious of those who seek to convince us with means other than reason, and of charismatic leaders: we must be cautious about delegating to others our judgment and our will. Since it is difficult to distinguish true prophets from false, it is as well to regard all prophets with suspicion. It is better to renounce revealed truths, even if they exalt us by their splendor or if we find them convenient because we can acquire them gratis. It is better to content oneself with other more modest and less exciting truths, those one acquires painfully, little by little and without shortcuts, with study, discussion, and reasoning, those that can be verified and demonstrated.

Three prisoners are brought to jail, but the jail is full. So the warden suggests the following procedure: Each of the three prisoners will have a colored hat placed on his head, either white or black or red. (Each color may appear any number of times — they may all get red hats, or two could be white and one black, etc.) Each prisoner will be able to see the others’ hats, but not his own. Without communicating with each other in any way, they must each write down on a piece of paper what color they think their own hat is. If at least one of them guesses correctly, they will all go free; otherwise, they will all be executed. The warden gives the prisoners a few minutes to talk it over before the procedure begins. What should they do to ensure their survival?

I like this puzzle because at first it seems impossible: there is no way for a prisoner to deduce anything about the color of his own hat based on the hats of the others, so what strategy could possibly ensure that at least one of them will guess correctly? The key is to think like a mathematician, and look at the underlying structure of the scenario. Try to solve it on your own before reading on.

Let us assign a number to each of the colors: zero for white, one for black, two for red. Now, consider the sum of the three numbers corresponding to the colors of the prisoners’ hats. We don’t know what this sum is, but each of the prisoners knows two out of its three components. Next, think of the remainder we will get if we divide that sum by three: it must be either zero or one or two. For example, if there are two red hats and one white hat, then the sum is four, and the remainder when dividing by three is one. The idea is for each of the prisoners to be assigned one of the possible remainders, and each will guess the color which, when added to the other two colors he sees, yields a sum whose remainder matches the one he was assigned.

According to the strategy described, in this instance prisoner #0 would guess black, prisoner #1 would guess red, and prisoner #2 would guess black.

For instance, if prisoners zero and one are wearing red hats, then prisoner two will guess that his hat is black (bringing the sum to five and the remainder to two). He could be wrong, of course — but exactly one of the prisoners will always be right: the one who was assigned the remainder that matches the actual sum of the three colors. In the above case, if prisoner two’s hat is white then prisoner one will guess correctly, and if prisoner two’s hat is red then prisoner zero will guess correctly.

In case you were worried, this strategy can naturally be generalized for n prisoners and n hat colors.